Here, \(\vec{\gamma _\ell } = (\gamma _{\ell ,a_1}, \gamma _{\ell ,a_2}, \ldots , \gamma _{\ell , a_m} ) \in \mathbb {R}^{m}\)  is the vector of angles used when \(U(\vec{\gamma _\ell }, C)\)  is applied for the \(\ell ^{th}\)  time and \(\vec{\beta _\ell } = (\beta _{\ell ,v_1}, \beta _{\ell ,v_2}, \ldots , \beta _{\ell , v_n}) \in \mathbb {R}^n\)  is the vector of \(\beta \)  angles for \(U(\vec{\beta _\ell }, B)\)  when applied for the \(\ell ^{th}\)  time. Each \(a\)  denotes a clause in the CO problem and \(v_j\)  refers to a specific qubit in the problem formulation, and \(B_v\)  typically refers to the Pauli-x matrix acting on qubit \(v\) . Each problem can be translated into a graph by identifying each qubit with a vertex and each interaction between qubits with an edge between the appropriate vertices [1]}, [2]}, [3]}.
